State Plane Coordinates vs. Surface Coordinates, Part 9.

This is the final article of this series. I know not everything was
covered, but it was a good review. I'm interested in comments because we may need to
have an additional article or two blended in with the GPS topics that will begin with the
next column.

To summarize, we converted the latitude and longitude coordinates,
determined by GPS, of stations BROMILOW, REILLY and WAKEMAN to state plane coordinates in
the New Mexico Central zone, using program SPCS83.

As can be seen from previous articles, it's a trivial task to work in
the state plane coordinate system. All slope distances are reduced to horizontal
distances, and these are multiplied by a combined factor to get grid distances. This is
the only thing different from working in a plane surface coordinate system. There is one
exception: If the traverse lines are long, a correction must be made to the azimuth.
Before I define "long line," let's look at Figure 1 (Figure 2.5 from NOAA
Manual NOS NGS 5, State Plane Coordinate System of 1983 by James E. Stem).

t=aÂ¿g+d Figure 1. Azimuths.

In Figure 1, the grid azimuths we calculated in our small traverse are
labeled t. The following is a quote from the NOAA manual:

The projection of the geodetic azimuth a onto the grid is not the grid
azimuth, but the projected geodetic azimuth symbolized as T. Convergence y is defined as
the difference between the geodetic and projected geodetic azimuths. Hence by definition,
a = T + y, and the sign of y should be applied accordingly. The angle obtained from two
projected geodetic azimuths is a true representation of an observed angle.

When an azimuth is computed from two plane coordinate
pairs, the resulting quantity is the grid azimuth symbolized as t. The relationship
between projected geodetic azimuth T and grid azimuth t is subtle and may be more clearly
understood in [Figure 1]. The difference between these azimuths is a computable quantity
symbolized as d or more often as (t-T). For the purpose of sign convention it is defined
as d= t-T. For reasons apparent in [Figure 1], this term is also identified as the
"arc-to-chord" correction.

Only in precise traverses is the (t-T) correction significant. An approximate equation is:
(t-T)=25.4(DN)(DE)10-10 where DN and DE are in meters. To solve this equation you will
need more information. For the Lambert projection, DN is the distance from the central
axis of the zone, and DE is the difference of eastings of the endpoints of the line. For
the Transverse Mercator projection, DE is the distance of the central axis of the
projection, and DN is the difference of northings of endpoints of the line. If you would
like to see an example of a traverse with long lines showing these calculations, let me
know.

Referring again to Figure 1, the angle y is the convergence angle (also
called the mapping angle). As can be seen, adding y to t gives the geodetic azimuth.
Referring to the last two columns, y (called convergence in the articles) is calculated
every time the program SPCS83 converts from geodetic coordinates to state plane
coordinates and from state plane coordinates to geodetic coordinates. If, after completing
a grid traverse, you want the geodetic azimuth, you add (algebraically) the convergence
angle to the grid azimuth. This is necessary because the meridians converge toward the
poles while the north-south grid lines are parallel to the central meridian. On the
central meridian of each zone, the convergence is zero.

Surface Coordinates

It was common to take each line on the grid and divide the length by the combined factor.
This gave the horizontal distance on the earthÂ¿s surface. Doing this allowed
conventional surveying crews to retrace grid lines. However, the use of computers in all
surveying offices has changed that. Today, surveyors want to convert the state plane
coordinates to surface coordinates (sometimes called project coordinates). I can tell you
that it's bad practice, but let's show how it is done.

There can be only one combined factor for a project if working in surface
coordinates. On anything other than short traverses, the scale factor changes for each
line, and the elevation factor may change. That's why state plane coordinates work.

Station
Name

Northing

Easting

Bromilow

142158.262m

452489.852m

Reilly

142268.912m

452506.387m

Wakeman

142399.023m

452131.948m

Â

Station
Name

Surface
Northing

Surface
Easting

Bromilow

2.158.262m

2489.852m

Reilly

2268.912m

2506.387m

Wakeman

2399.023m

2131.948m

So that there is no confusion, it's a common practice
to drop the first digit or two from the station's state plane coordinate. In our
case, the state plane coordinates of the three known stations were as follows:

A person wanting surface coordinates might do the following:

(But in most cases the coordinates would be in feet.)

Let me tell of two different situations where surface coordinates were
used and problems detected. Two civil engineering firms were given five miles of highway
design by a state department of transportation; the projects were end-to-end. Both firms
used surface coordinates. When the first firm finished its work, its ending coordinates
did not match the beginning coordinates of the second firm's project.

In another incident, I received an E-mail asking if I could help a
surveyor having problems with his state plane coordinates. When I called him, he said the
problem was not the state plane coordinates, but his surface coordinates, which were only
good to about 10 miles. He was looking for a way to extend beyond 10 miles.

I need to talk to more people who use surface coordinates. With only one
combined factor per project, this can create havoc. I hope you can see from this series of
articles that state plane coordinates are the way to go.

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